Chapter 13 of the PDF Expert Guide covers syncing. It should have downloaded as a part of PDF Expert and be readable within the app.
Basically you go to Networks, add your DropBox account if necessary, browse to the folder you want to sync, then tap the Sync button at the top right. In Settings make sure Sync is set to either Wi-Fi or Wi-Fi and cellular; otherwise the app only syncs manually, when you tap the Sync button.
If the files are already in local folder in PDF Expert, you will have to move them to the selected DropBox folder(s). You can do this by choosing the Documents view, navigating to the folder/files you want to move, tapping the Edit button at the top right, checking the files you wish to move, then using the Move tool.
In my experience changes made to a document sync back to DropBox quickly. Occasionally I will have to force a sync (by tapping the Sync button) to get documents to sync from DropBox to PDF Expert; but that could be impatience on my part. The app should sync on opening, and every 10 minutes after that. It will not sync if the app is closed, though that behavior may have changed somewhat since iOS7, if the developer enabled background updates.